Powers of 2 with no Zero in Decimal Representation

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Unproven Hypotheses

The following powers of $2$ which contain no zero in their decimal representation are believed to be all that exist:

$1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 8192, 16 \, 384,$
$32 \, 768, 65 \, 536, 262 \, 144, 524 \, 288, 16 \, 777 \, 216, 33 \, 554 \, 432, 134 \, 217 \, 728,$
$268 \, 435 \, 456, 2 \, 147 \, 483 \, 648, 4 \, 294 \, 967 \, 296, 8 \, 589 \, 934 \, 592, 17 \, 179 \, 869 \, 184,$
$34 \, 359 \, 738 \, 368, 68 \, 719 \, 476 \, 736, 137 \, 438 \, 953 \, 472, 549 \, 755 \, 813 \, 888,$
$562 \, 949 \, 953 \, 421 \, 312, 2 \, 251 \, 799 \, 813 \, 685 \, 248, 147 \, 573 \, 952 \, 589 \, 676 \, 412 \, 928,$
$4 \, 722 \, 366 \, 482 \, 869 \, 645 \, 213 \, 696, 75 \, 557 \, 863 \, 725 \, 914 \, 323 \, 419 \, 136, 151 \, 115 \, 727 \, 451 \, 828 \, 646 \, 838 \, 272,$
$2 \, 417 \, 851 \, 639 \, 229 \, 258 \, 349 \, 412 \, 352, 77 \, 371 \, 252 \, 455 \, 336 \, 267 \, 181 \, 195 \, 264$

but this has not been proven.

This sequence is A238938 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


The corresponding indices are:

$0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15, 16, 18, 19, 24, 25, 27, 28, 31, 32, 33, 34, 35, 36, 37, 39, 49, 51, 67, 72, 76, 77, 81, 86$

This sequence is A007377 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Progress

\(\ds 2^0\) \(=\) \(\ds 1\)
\(\ds 2^1\) \(=\) \(\ds 2\)
\(\ds 2^2\) \(=\) \(\ds 4\)
\(\ds 2^3\) \(=\) \(\ds 8\)
\(\ds 2^4\) \(=\) \(\ds 16\)
\(\ds 2^5\) \(=\) \(\ds 32\)
\(\ds 2^6\) \(=\) \(\ds 64\)
\(\ds 2^7\) \(=\) \(\ds 128\)
\(\ds 2^9\) \(=\) \(\ds 512\)
\(\ds 2^{10}\) \(=\) \(\ds 1024\) which contains a zero
\(\ds 2^{11}\) \(=\) \(\ds 2048\) which contains a zero
\(\ds 2^{12}\) \(=\) \(\ds 4096\) which contains a zero
\(\ds 2^{13}\) \(=\) \(\ds 8192\)
\(\ds 2^{14}\) \(=\) \(\ds 16 \, 384\)
\(\ds 2^{15}\) \(=\) \(\ds 32 \, 768\)
\(\ds 2^{16}\) \(=\) \(\ds 65 \, 536\)
\(\ds 2^{17}\) \(=\) \(\ds 131 \, 072\) which contains a zero
\(\ds 2^{18}\) \(=\) \(\ds 262 \, 144\)
\(\ds 2^{19}\) \(=\) \(\ds 524 \, 288\)
\(\ds 2^{20}\) \(=\) \(\ds 1 \, 048 \, 576\) which contains a zero
\(\ds 2^{21}\) \(=\) \(\ds 2 \, 097 \, 152\) which contains a zero
\(\ds 2^{22}\) \(=\) \(\ds 4 \, 194 \, 304\) which contains a zero
\(\ds 2^{23}\) \(=\) \(\ds 8 \, 388 \, 608\) which contains a zero
\(\ds 2^{24}\) \(=\) \(\ds 16 \, 777 \, 216\)
\(\ds 2^{25}\) \(=\) \(\ds 33 \, 554 \, 432\)
\(\ds 2^{26}\) \(=\) \(\ds 67 \, 108 \, 864\) which contains a zero
\(\ds 2^{27}\) \(=\) \(\ds 134 \, 217 \, 728\)
\(\ds 2^{28}\) \(=\) \(\ds 268 \, 435 \, 456\)
\(\ds 2^{29}\) \(=\) \(\ds 536 \, 870 \, 912\) which contains a zero
\(\ds 2^{30}\) \(=\) \(\ds 1 \, 073 \, 741 \, 824\) which contains a zero
\(\ds 2^{31}\) \(=\) \(\ds 2 \, 147 \, 483 \, 648\)
\(\ds 2^{32}\) \(=\) \(\ds 4 \, 294 \, 967 \, 296\)
\(\ds 2^{33}\) \(=\) \(\ds 8 \, 589 \, 934 \, 592\)
\(\ds 2^{34}\) \(=\) \(\ds 17 \, 179 \, 869 \, 184\)
\(\ds 2^{35}\) \(=\) \(\ds 34 \, 359 \, 738 \, 368\)
\(\ds 2^{36}\) \(=\) \(\ds 68 \, 719 \, 476 \, 736\)
\(\ds 2^{37}\) \(=\) \(\ds 137 \, 438 \, 953 \, 472\)
\(\ds 2^{38}\) \(=\) \(\ds 274 \, 877 \, 906 \, 944\) which contains a zero
\(\ds 2^{39}\) \(=\) \(\ds 549 \, 755 \, 813 \, 888\)
\(\ds 2^{40}\) \(=\) \(\ds 1 \, 099 \, 511 \, 627 \, 776\) which contains a zero
\(\ds 2^{41}\) \(=\) \(\ds 2 \, 199 \, 023 \, 255 \, 552\) which contains a zero
\(\ds 2^{42}\) \(=\) \(\ds 4 \, 398 \, 046 \, 511 \, 104\) which contains zeroes
\(\ds 2^{43}\) \(=\) \(\ds 8 \, 796 \, 093 \, 022 \, 208\) which contains zeroes
\(\ds 2^{44}\) \(=\) \(\ds 17 \, 592 \, 186 \, 044 \, 416\) which contains a zero
\(\ds 2^{45}\) \(=\) \(\ds 35 \, 184 \, 372 \, 088 \, 832\) which contains a zero
\(\ds 2^{46}\) \(=\) \(\ds 70 \, 368 \, 744 \, 177 \, 664\) which contains a zero
\(\ds 2^{47}\) \(=\) \(\ds 140 \, 737 \, 488 \, 355 \, 328\) which contains a zero
\(\ds 2^{48}\) \(=\) \(\ds 281 \, 474 \, 976 \, 710 \, 656\) which contains a zero
\(\ds 2^{49}\) \(=\) \(\ds 562 \, 949 \, 953 \, 421 \, 312\)
\(\ds 2^{50}\) \(=\) \(\ds 1 \, 125 \, 899 \, 906 \, 842 \, 624\) which contains a zero
\(\ds 2^{51}\) \(=\) \(\ds 2 \, 251 \, 799 \, 813 \, 685 \, 248\)
\(\ds 2^{52}\) \(=\) \(\ds 4 \, 503 \, 599 \, 627 \, 370 \, 496\) which contains zeroes
\(\ds 2^{53}\) \(=\) \(\ds 9 \, 007 \, 199 \, 254 \, 740 \, 992\) which contains zeroes
\(\ds 2^{54}\) \(=\) \(\ds 18 \, 014 \, 398 \, 509 \, 481 \, 984\) which contains a zero
\(\ds 2^{55}\) \(=\) \(\ds 36 \, 028 \, 797 \, 018 \, 963 \, 968\) which contains zeroes
\(\ds 2^{56}\) \(=\) \(\ds 72 \, 057 \, 594 \, 037 \, 927 \, 936\) which contains zeroes
\(\ds 2^{57}\) \(=\) \(\ds 144 \, 115 \, 188 \, 075 \, 855 \, 872\) which contains a zero
\(\ds 2^{58}\) \(=\) \(\ds 288 \, 230 \, 376 \, 151 \, 711 \, 744\) which contains a zero
\(\ds 2^{59}\) \(=\) \(\ds 576 \, 460 \, 752 \, 303 \, 423 \, 488\) which contains zeroes
\(\ds 2^{60}\) \(=\) \(\ds 1 \, 152 \, 921 \, 504 \, 606 \, 846 \, 976\) which contains zeroes
\(\ds 2^{61}\) \(=\) \(\ds 2 \, 305 \, 843 \, 009 \, 213 \, 693 \, 952\) which contains zeroes
\(\ds 2^{62}\) \(=\) \(\ds 4 \, 611 \, 686 \, 018 \, 427 \, 387 \, 904\) which contains zeroes
\(\ds 2^{63}\) \(=\) \(\ds 9 \, 223 \, 372 \, 036 \, 854 \, 775 \, 808\) which contains a zero
\(\ds 2^{64}\) \(=\) \(\ds 18 \, 446 \, 744 \, 073 \, 709 \, 551 \, 616\) which contains zeroes
\(\ds 2^{65}\) \(=\) \(\ds 36 \, 893 \, 488 \, 147 \, 419 \, 103 \, 232\) which contains a zero
\(\ds 2^{66}\) \(=\) \(\ds 73 \, 786 \, 976 \, 294 \, 838 \, 206 \, 464\) which contains a zero
\(\ds 2^{67}\) \(=\) \(\ds 147 \, 573 \, 952 \, 589 \, 676 \, 412 \, 928\)
\(\ds 2^{68}\) \(=\) \(\ds 295 \, 147 \, 905 \, 179 \, 352 \, 825 \, 856\) which contains a zero
\(\ds 2^{69}\) \(=\) \(\ds 590 \, 295 \, 810 \, 358 \, 705 \, 651 \, 712\) which contains zeroes
\(\ds 2^{70}\) \(=\) \(\ds 1 \, 180 \, 591 \, 620 \, 717 \, 411 \, 303 \, 424\) which contains zeroes
\(\ds 2^{71}\) \(=\) \(\ds 2 \, 361 \, 183 \, 241 \, 434 \, 822 \, 606 \, 848\) which contains a zero
\(\ds 2^{72}\) \(=\) \(\ds 4 \, 722 \, 366 \, 482 \, 869 \, 645 \, 213 \, 696\)
\(\ds 2^{73}\) \(=\) \(\ds 9 \, 444 \, 732 \, 965 \, 739 \, 290 \, 427 \, 392\) which contains a zero
\(\ds 2^{74}\) \(=\) \(\ds 18 \, 889 \, 465 \, 931 \, 478 \, 580 \, 854 \, 784\) which contains a zero
\(\ds 2^{75}\) \(=\) \(\ds 37 \, 778 \, 931 \, 862 \, 957 \, 161 \, 709 \, 568\) which contains a zero
\(\ds 2^{76}\) \(=\) \(\ds 75 \, 557 \, 863 \, 725 \, 914 \, 323 \, 419 \, 136\)
\(\ds 2^{77}\) \(=\) \(\ds 151 \, 115 \, 727 \, 451 \, 828 \, 646 \, 838 \, 272\)
\(\ds 2^{78}\) \(=\) \(\ds 302 \, 231 \, 454 \, 903 \, 657 \, 293 \, 676 \, 544\) which contains zeroes
\(\ds 2^{79}\) \(=\) \(\ds 604 \, 462 \, 909 \, 807 \, 314 \, 587 \, 353 \, 088\) which contains zeroes
\(\ds 2^{80}\) \(=\) \(\ds 1 \, 208 \, 925 \, 819 \, 614 \, 629 \, 174 \, 706 \, 176\) which contains zeroes
\(\ds 2^{81}\) \(=\) \(\ds 2 \, 417 \, 851 \, 639 \, 229 \, 258 \, 349 \, 412 \, 352\)
\(\ds 2^{82}\) \(=\) \(\ds 4 \, 835 \, 703 \, 278 \, 458 \, 516 \, 698 \, 824 \, 704\) which contains zeroes
\(\ds 2^{83}\) \(=\) \(\ds 9 \, 671 \, 406 \, 556 \, 917 \, 033 \, 397 \, 649 \, 408\) which contains zeroes
\(\ds 2^{84}\) \(=\) \(\ds 19 \, 342 \, 813 \, 113 \, 834 \, 066 \, 795 \, 298 \, 816\) which contains a zero
\(\ds 2^{85}\) \(=\) \(\ds 38 \, 685 \, 626 \, 227 \, 668 \, 133 \, 590 \, 597 \, 632\) which contains a zero
\(\ds 2^{86}\) \(=\) \(\ds 77 \, 371 \, 252 \, 455 \, 336 \, 267 \, 181 \, 195 \, 264\)

$\blacksquare$


Sources